Chapter 11.2: The Five-Phase Diagram

From Three Phases to Five

The pure TASEP has three phases (LD, HD, MC). When Langmuir kinetics are added, the competition between boundary-driven transport and bulk attachment/detachment produces two additional phases: an LD-HD coexistence phase and an LD-MC-HD triple coexistence phase near a multicritical point. The Langmuir equilibrium density$\rho_{\text{eq}}$ acts as a pinning field that localises domain walls at specific positions.

11.2.1 Identifying the Five Phases

Phase I: Low Density (LD)

The entire lattice is in a low-density state controlled by the left boundary. The density profile smoothly approaches $\rho_{\text{eq}}$from below but never reaches the high-density branch:

$$\rho(x) < \frac{1}{2} \quad \forall\, x, \qquad J = \alpha(1-\alpha) + O(\Omega)$$

Phase II: High Density (HD)

The entire lattice is in a high-density state controlled by the right boundary:

$$\rho(x) > \frac{1}{2} \quad \forall\, x, \qquad J = \beta(1-\beta) + O(\Omega)$$

Phase III: Maximal Current (MC)

Both boundaries are fast enough that the bulk self-organises to maximise throughput:

$$\rho \approx \frac{1}{2}, \qquad J \approx \frac{1}{4}$$

Phase IV: LD-HD Coexistence

A domain wall separates a low-density left region from a high-density right region. Unlike the pure TASEP coexistence line (where the wall delocalises), Langmuir kinetics pin the wall at a specific position$x_w$:

$$x_w = \frac{\rho_{\text{eq}} - \rho_-}{\rho_+ - \rho_-}$$

where $\rho_- = \alpha$ (LD branch) and$\rho_+ = 1 - \beta$ (HD branch).

Phase V: LD-MC-HD Triple Coexistence

Near the multicritical point, a region of maximal current appears between the LD and HD domains. This occurs when $\rho_{\text{eq}} \approx 1/2$and $\alpha, \beta$ are close to$1/2$. Three regions coexist with two domain walls.

11.2.2 Domain Wall Theory

The domain wall position $x_w$ can be understood as a particle performing biased diffusion in an effective potential. Define the domain wall as the interface between densities$\rho_-$ and$\rho_+$.

Effective Potential

The global continuity equation, integrated over the system, gives a condition on the wall position. The Langmuir terms create an effective potential:

$$V(x_w) = -\Omega \int_0^{x_w} \bigl[\omega_A(1-\rho_-(x)) - \omega_D \rho_-(x)\bigr] dx - \Omega \int_{x_w}^{1} \bigl[\omega_A(1-\rho_+(x)) - \omega_D \rho_+(x)\bigr] dx$$

The equilibrium wall position minimises this potential. The curvature of$V(x_w)$ at the minimum determines the wall fluctuations:

$$D_{\text{wall}} \sim \frac{1}{L(\rho_+ - \rho_-)^2}$$

The wall diffusion coefficient scales as$1/L$, so wall fluctuations are$O(1/\sqrt{L})$—the wall islocalised in the thermodynamic limit. This is the key difference from the pure TASEP, where the wall is delocalised on the coexistence line.

Derivation from Global Continuity

In steady state, the total current must balance the total Langmuir source. Integrating the continuity equation over the full system:

$$J_{\text{out}} - J_{\text{in}} = \int_0^L \bigl[\Omega_A(1-\rho) - \Omega_D \rho\bigr] dx$$

Substituting the piecewise density profile (LD for$x < x_w$, HD for$x > x_w$) and solving for$x_w$ gives the wall position formula.

11.2.3 Spectral Gap and KPZ-to-Diffusive Crossover

For the general ASEP (without Langmuir), the Bethe ansatz gives the exact spectral gap of the Markov generator:

$$\Delta = p + q - 2\sqrt{pq}\cos\!\left(\frac{\pi}{L}\right) \approx \frac{\pi^2 \sqrt{pq}}{L^2} \quad (L \gg 1)$$

This gives a relaxation time $\tau \sim L^2$(diffusive) for the SSEP ($p = q$). For the TASEP ($q = 0$), the spectral gap is$\Delta \sim L^{-3/2}$ (KPZ). The full crossover is:

$$\tau \sim \begin{cases} L^{3/2} & q \to 0 \quad \text{(KPZ)} \\ L^2 & q \to p \quad \text{(diffusive)} \end{cases}$$

Adding Langmuir kinetics with $\Omega \sim 1$introduces a new length scale $\ell_{\Omega} = 1/\Omega$. For $L \gg \ell_{\Omega}$, the relaxation becomes exponentially fast as the Langmuir kinetics provide a local restoring force.

11.2.4 Simulation: Five-Phase Diagram and Domain Walls

Five-Phase Diagram: ASEP + Langmuir

Python

script.py109 lines

import numpy as np
import matplotlib.pyplot as plt
import matplotlib
matplotlib.use('Agg')

# ── Five-Phase Diagram for ASEP + Langmuir ──
# We map the phase diagram in the (alpha, beta) plane for fixed Omega

fig, axes = plt.subplots(1, 3, figsize=(16, 5.5))
fig.patch.set_facecolor('#0a0a0a')

Omega_A_vals = [0.5, 0.5, 0.8]
Omega_D_vals = [0.5, 0.3, 0.2]
titles = ['Ω_A=Ω_D=0.5 (ρ_eq=0.5)', 'Ω_A=0.5, Ω_D=0.3 (ρ_eq=0.625)', 'Ω_A=0.8, Ω_D=0.2 (ρ_eq=0.8)']

for panel, (OA, OD, title) in enumerate(zip(Omega_A_vals, Omega_D_vals, titles)):
    ax = axes[panel]
    ax.set_facecolor('#0a0a0a')
    ax.tick_params(colors='white')
    for spine in ax.spines.values():
        spine.set_color('#334155')

rho_eq = OA / (OA + OD)
    N = 300
    alpha_arr = np.linspace(0.01, 0.99, N)
    beta_arr = np.linspace(0.01, 0.99, N)
    AA, BB = np.meshgrid(alpha_arr, beta_arr)

# Phase classification
    phase = np.zeros_like(AA)

for i in range(N):
        for j in range(N):
            a = AA[i, j]
            b = BB[i, j]
            rho_minus = a       # LD density
            rho_plus = 1 - b    # HD density

# Wall position
            if rho_plus > rho_minus:
                x_w = (rho_eq - rho_minus) / (rho_plus - rho_minus)
            else:
                x_w = 0.5

if a >= 0.5 and b >= 0.5:
                phase[i, j] = 3  # MC
            elif a < 0.5 and b >= 0.5 and a < b:
                if rho_eq <= a:
                    phase[i, j] = 1  # Pure LD
                else:
                    phase[i, j] = 1  # LD (Langmuir pushes toward eq but boundary dominates)
            elif b < 0.5 and a >= 0.5 and b < a:
                if rho_eq >= 1 - b:
                    phase[i, j] = 2  # Pure HD
                else:
                    phase[i, j] = 2
            elif a < 0.5 and b < 0.5:
                if x_w < 0:
                    phase[i, j] = 1  # LD
                elif x_w > 1:
                    phase[i, j] = 2  # HD
                elif 0 < x_w < 1:
                    phase[i, j] = 4  # LD-HD coexistence
                elif a == b:
                    phase[i, j] = 4
                else:
                    if a < b:
                        phase[i, j] = 1
                    else:
                        phase[i, j] = 2
            else:
                # Near multicritical point
                if abs(rho_eq - 0.5) < 0.1 and abs(a - 0.5) < 0.1 and abs(b - 0.5) < 0.1:
                    phase[i, j] = 5  # Triple coexistence
                elif a < b:
                    phase[i, j] = 1
                else:
                    phase[i, j] = 2

from matplotlib.colors import ListedColormap
    cmap5 = ListedColormap(['#0a0a0a', '#065f46', '#7f1d1d', '#1e3a5f', '#4a1d7f', '#7f4a1d'])
    ax.pcolormesh(AA, BB, phase, cmap=cmap5, shading='auto', alpha=0.5, vmin=0, vmax=5)

# Phase boundaries
    ax.plot([0, 0.5], [0, 0.5], color='#fbbf24', linewidth=2.5, label='LD/HD')
    ax.plot([0.5, 0.5], [0.5, 1.0], color='#34d399', linewidth=2.5, label='LD/MC')
    ax.plot([0.5, 1.0], [0.5, 0.5], color='#f472b6', linewidth=2.5, label='HD/MC')

# Labels
    ax.text(0.2, 0.7, 'LD', fontsize=16, color='#34d399', fontweight='bold', ha='center')
    ax.text(0.7, 0.2, 'HD', fontsize=16, color='#f87171', fontweight='bold', ha='center')
    ax.text(0.75, 0.75, 'MC', fontsize=16, color='#60a5fa', fontweight='bold', ha='center')
    if rho_eq > 0.2 and rho_eq < 0.8:
        ax.text(0.25, 0.25, 'LD+HD', fontsize=11, color='#c084fc', fontweight='bold', ha='center')

ax.set_xlabel('α', color='white', fontsize=12)
    ax.set_ylabel('β', color='white', fontsize=12)
    ax.set_title(title, color='#6ee7b7', fontsize=11)
    ax.set_xlim(0, 1)
    ax.set_ylim(0, 1)
    ax.set_aspect('equal')
    if panel == 0:
        ax.legend(facecolor='#1a1a2e', edgecolor='#334155', labelcolor='white', fontsize=8)

plt.tight_layout(pad=2.0)
plt.savefig('output.png', dpi=150, bbox_inches='tight', facecolor='#0a0a0a')
plt.close()
print("Five-phase diagrams for different Langmuir parameters plotted.")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

Domain Wall Position vs Langmuir Parameter

Python

script.py71 lines

import numpy as np
import matplotlib.pyplot as plt
import matplotlib
matplotlib.use('Agg')

# ── Domain wall position as a function of Omega ──
fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(12, 5))
fig.patch.set_facecolor('#0a0a0a')

for ax in [ax1, ax2]:
    ax.set_facecolor('#0a0a0a')
    ax.tick_params(colors='white')
    for spine in ax.spines.values():
        spine.set_color('#334155')

# Fixed boundary rates
alpha = 0.3
beta = 0.3
rho_minus = alpha
rho_plus = 1 - beta

# Vary Omega_A/Omega_D ratio
K_vals = np.linspace(0.1, 5.0, 200)
x_w_vals = []
for K in K_vals:
    rho_eq = K / (1 + K)
    x_w = (rho_eq - rho_minus) / (rho_plus - rho_minus)
    x_w_vals.append(np.clip(x_w, 0, 1))

ax1.plot(K_vals, x_w_vals, color='#34d399', linewidth=2.5)
ax1.axhline(y=0.5, color='#fbbf24', linestyle='--', alpha=0.5, label='x_w = 0.5')
ax1.axvline(x=1.0, color='#f472b6', linestyle=':', alpha=0.5, label='K=1 (ρ_eq=0.5)')
ax1.fill_between(K_vals, 0, x_w_vals, alpha=0.1, color='#34d399')
ax1.set_xlabel('Langmuir constant K = Ω_A/Ω_D', color='white')
ax1.set_ylabel('Wall position x_w/L', color='white')
ax1.set_title(f'Domain Wall Position (α=β={alpha})', color='#6ee7b7', fontsize=13)
ax1.legend(facecolor='#1a1a2e', edgecolor='#334155', labelcolor='white')
ax1.set_ylim(-0.05, 1.05)

# Domain wall profiles for different Omega
Omega_total = [0.5, 1.0, 2.0, 5.0]
colors_dw = ['#34d399', '#2dd4bf', '#fbbf24', '#f87171']
x_arr = np.linspace(0, 1, 500)

for i, Omega in enumerate(Omega_total):
    Omega_A = Omega / 2
    Omega_D = Omega / 2
    rho_eq = 0.5
    x_w = (rho_eq - alpha) / (1 - beta - alpha)

# Approximate density profile with smoothed domain wall
    width = 0.05 / np.sqrt(Omega + 0.1)  # Wall sharpens with increasing Omega
    profile = rho_minus + (rho_plus - rho_minus) / (1 + np.exp(-(x_arr - x_w) / width))

ax2.plot(x_arr, profile, color=colors_dw[i], linewidth=2,
            label=f'Ω = {Omega}')

ax2.axhline(y=0.5, color='white', linestyle='--', alpha=0.3)
ax2.axhline(y=alpha, color='#34d399', linestyle=':', alpha=0.3, label=f'ρ_- = {alpha}')
ax2.axhline(y=1-beta, color='#f87171', linestyle=':', alpha=0.3, label=f'ρ_+ = {1-beta}')
ax2.set_xlabel('Position x/L', color='white')
ax2.set_ylabel('Density ρ', color='white')
ax2.set_title('Domain Wall Profiles', color='#6ee7b7', fontsize=13)
ax2.legend(facecolor='#1a1a2e', edgecolor='#334155', labelcolor='white', fontsize=9)
ax2.set_ylim(-0.05, 1.05)

plt.tight_layout(pad=2.0)
plt.savefig('output.png', dpi=150, bbox_inches='tight', facecolor='#0a0a0a')
plt.close()
print("Domain wall position and profile analysis complete.")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

Key Takeaways

ASEP + Langmuir kinetics produces a five-phase diagram: LD, HD, MC, LD-HD coexistence, and LD-MC-HD triple coexistence.
The domain wall position is$x_w = (\rho_{\text{eq}} - \rho_-)/(\rho_+ - \rho_-)$, pinned by the Langmuir equilibrium density.
Wall fluctuations scale as $O(1/\sqrt{L})$, meaning the wall is localised in the thermodynamic limit.
The spectral gap crosses over from KPZ ($\tau \sim L^{3/2}$) to diffusive ($\tau \sim L^2$) as$q/p$ increases.
In traffic terms: side-street access (Langmuir) pins traffic jams at predictable locations, unlike boundary-only models where jam position fluctuates.

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